Outside of the Atlantic and Med the Soviet Navy would constitute a nuisance requiring attention but would end up like the German Asiatic Squadron in WWI - if lucky make a splash do some damage, but eventually go down. In the Med, the issue would be how much damage the Soviet Navy could do to NATO navies before they moored on the bottom. To the extent any surface ships of any size survived the Soviet Navy would control the Black Sea, and at least the Eastern Baltic - I doubt any NATO surface vessels other than small combatants belonging to Germany/Denmark would be seen east of Denmark. The key areas would be the Atlantic, whatever other naval actions went on around the world. Heavy equipment for the US forces in Europe comes almost exclusively by sea, so the basic Soviet plan was to close this pipeline down as much as possible for long enough to allow the land forces to achieve their goals. Obviously the more warning/buildup there is to the war, this works in favor of NATO as they can ship freely until the war starts, it does allow the Soviets to surge their submarines which will be detected and comprise a war warning.
Obviously there were other missions for the Soviet Navy, although their capabilities in the 60s were more limited than in the late 70s and 80s - supporting amphibious assaults in Norway and the Baltic is one example. Since the few boomers they had in the 60s had short range (relatively) missiles, establishing and protecting an SSBN bastion was not a mission yet. If we a saying the war remains entirely conventional, then the "close the Atlantic bridge" mission becomes even more critical for both sides, as conventional reinforcement of NATO by the USA, if slowed enough, means that the option of replacing tanks and infantry with tactical nukes is not there.